2018 HCI P2 Q10

Each morning, Tony drives from his home to his office and has to pass \(4\) traffic lights on his way. He has to reach his office by 8.50 am. The driving time to his office and the time held up at a traffic light junction, in minutes, may be assumed to follow normal distributions with means and standard deviations as summarized below:

Tony leaves home at 8.30 am. If the probability that Tony is not late is \(0.713\), show that \(\mu =1.2\), correct to \(1\) decimal place. State an assumption needed in your calculations.[4]

1. For \(10\) mornings, Tony leaves home at 8.30 am. Find the probability that he arrives late at his office for the third time on the \({{10}^{\mathrm{th}}}\) day.[2]
2. Find the probability that Tony’s driving time to his office is less than \(10\) times the time he is held up at a traffic light junction.[2]
3. Find the probability that Tony’s driving time to his office and the total time he is held up at the \(4\) traffic light junctions differs by more than \(8\) minutes.[3]

Assume \(\mu \) is unknown.

4. The time held up at a traffic light junction is recorded on n randomly chosen occasions. Find the smallest \(n\) so that it is at least \(98\%\) certain that the sample mean time Tony is held up at a traffic light junction is within \(5\) seconds of \(\mu \).[3]